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<h1>Pair Correlations</h1>

<h2>General Pair Correlation Function</h2>
<p>
The pair correlation function (PCF) between two species
α and β is
<div class="math">
g<sub>αβ</sub>(<b>r</b>, <b>r</b>')
= 〈n<sub>α</sub>(<b>r</b>)n<sub>β</sub>(<b>r</b>')〉/
  &nbsp;〈n(<b>r</b>)〉〈n(<b>r</b>')〉.
</div>
When the density fluctations are uncorrelated at large distances, such as
in a liquid, then <span class="math">g → 1</span>.
</p>

<h2>Pair Correlations in an Isotropic, Homegeneous System</h2>
<p>For an isotropic, homegenous system, the PCF becomes a 
function of <span class="math">r = |<b>r</b> - <b>r</b>'|</span>,
the distance between 
<span class="math"><b>r</b></span> and <span class="math"><b>r</b>'</span>.
</p>

<h2>Static Structure Factor</h2>
<a name="sk"/>
<p>
For a single component with <span class="math">N</span>
particles in the periodic cell, the static structure factor is
<div class="math">
S(<b>k</b>) = (1/N) 〈n<sub><b>k</b></sub>n<sub><b>-k</b></sub>〉,
</div>
where <span class="math">n<sub><b>k</b></sub> = Σ<sub>j</sub>
e<sup>-i<b>k</b>·<b>r</b><sub>j</sub></sup></span>.
Thus <span class="math">S(<b>k</b>)</span> is related to 
<span class="math">g(<b>r</b>, <b>r</b>')</span> by
<div class="math">
S(<b>k</b>) = ∫e<sup>-i<b>k</b>·(<b>r</b>-<b>r</b>')</sup> 
 [g(<b>r</b>,<b>r</b>')-1] d<b>r</b>.
</div>
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